It was shown by [1]}, [2]} that if a synchronous game has a perfect strategy (i.e. a strategy that wins with probability 1) then it also has a perfect synchronous strategy. Furthermore, if the former strategy is finite-dimensional, then so is the latter synchronous strategy. This characterization of perfect (finite-dimensional) strategies for synchronous games was recently extended to the case of near-perfect strategies in [3]}: any finite-dimensional strategy for a synchronous game \(G\)  that succeeds with high probability can be well-approximated by a convex combination of finite-dimensional synchronous strategies. In particular, results such as Theorem REF  for synchronous strategies extend in a natural way to general strategies.
